| L(s) = 1 | − 4-s + 2·5-s − 9-s + 4·11-s + 16-s + 12·19-s − 2·20-s − 25-s − 12·29-s − 4·31-s + 36-s + 4·41-s − 4·44-s − 2·45-s − 49-s + 8·55-s + 16·59-s − 20·61-s − 64-s − 12·71-s − 12·76-s + 24·79-s + 2·80-s + 81-s + 20·89-s + 24·95-s − 4·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 2.75·19-s − 0.447·20-s − 1/5·25-s − 2.22·29-s − 0.718·31-s + 1/6·36-s + 0.624·41-s − 0.603·44-s − 0.298·45-s − 1/7·49-s + 1.07·55-s + 2.08·59-s − 2.56·61-s − 1/8·64-s − 1.42·71-s − 1.37·76-s + 2.70·79-s + 0.223·80-s + 1/9·81-s + 2.11·89-s + 2.46·95-s − 0.402·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486889448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486889448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67049834252611502316942207888, −12.04923802423061998963216648099, −11.50660462668572456523772294422, −11.46241751778221374578123629196, −10.57939948210151857012249908720, −10.09032348670532363079607383801, −9.421221370828641681268391980188, −9.197074951567093821059782080266, −9.118498699472213274919546382730, −8.085221601482870724138762015961, −7.38038611945640380530025546703, −7.31401588725773932519732375859, −6.16253960391928455211864751269, −5.93081486344308861734746911120, −5.28833018134119284187123702292, −4.79178130173635207949765314522, −3.60825502339670272635639914580, −3.50514779740060973345442696270, −2.19614907503391920152352820609, −1.23492869683336486031404324122,
1.23492869683336486031404324122, 2.19614907503391920152352820609, 3.50514779740060973345442696270, 3.60825502339670272635639914580, 4.79178130173635207949765314522, 5.28833018134119284187123702292, 5.93081486344308861734746911120, 6.16253960391928455211864751269, 7.31401588725773932519732375859, 7.38038611945640380530025546703, 8.085221601482870724138762015961, 9.118498699472213274919546382730, 9.197074951567093821059782080266, 9.421221370828641681268391980188, 10.09032348670532363079607383801, 10.57939948210151857012249908720, 11.46241751778221374578123629196, 11.50660462668572456523772294422, 12.04923802423061998963216648099, 12.67049834252611502316942207888
